优化 / 数学规划
从一个科学解的集合中,寻找出最优元素
优化问题的通用形式
minimize f0(x)f_0(x)f0(x)
subject to fi(x)≤bi,i=1,...,Mf_i(x) \leq b_i ,i = 1,...,Mfi(x)≤bi,i=1,...,M
优化变量(optimization variable): x=[x1,...,xn]Tx = [x_1,...,x_n]^Tx=[x1,...,xn]T
目标函数(objective function): f0:Rn→Rf_0: R^n \rightarrow Rf0:Rn→R
可行解集(feasible set): { fi(z)≤bi,i=1,...,Mf_i(z) \leq b_i,i=1,...,Mfi(z)≤bi,i=1,...,M }
最优解:x∗⇔∀z,z∈{fi(z)≤bi,i=1,...,M},f0(z)⩾f0(x∗)x^* \Leftrightarrow \forall z, z \in \left \{ f_i(z) \leq b_i,i=1,...,M \right \} , f_0(z) \geqslant f_0(x^*)x∗⇔∀z,z∈{fi(z)≤bi,i=1,...,M},f0(z)⩾f0(x∗)
优化问题分类
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分类方法1:线性规划 / 非线性规划
fi(αx+βy)=αfi(x)+βfi(y),i=0,1,...,Mf_i(\alpha x + \beta y) = \alpha f_i(x) + \beta f_i(y) , i = 0,1,...,Mfi(αx+βy)=αfi(x)+βfi(y),i=0,1,...,M -
凸优化/非凸优化
fi(αx+βy)≤αfi(x)+βfi(y),i=0,1,...,Mf_i(\alpha x + \beta y) \leq \alpha f_i(x) + \beta f_i(y) , i = 0,1,...,Mfi(αx+βy)≤αfi(x)+βfi(y),i=0,1,...,M -
光滑 / 非光滑(目标函数)
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连续 / 离散(可行域)
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单目标 / 多目标